Persistence for a class of triangular cross diffusion parabolic systems

Advanced Nonlinear Studies (2005), 509–530 Persistence for a Class of Triangular Cross Diffusion Parabolic Systems Dung∗ Le Department of Applied Mathematics, University of Texas, San Antonio, Texas 78249 e-mail: dle@math.utsa.edu Toan Trong Nguyen Department of Applied Mathematics, University of Texas, San Antonio, Texas 78249 e-mail: toan.nguyen@utsa.edu Received 14 July 2005 Communicated by Chris Cosner Abstract The purpose of this paper is to investigate the dynamics of a class of triangular parabolic systems given on bounded domains of arbitrary dimension In particular, the existence of global attractors and the persistence property will be established 1991 Mathematics Subject Classification 35K65, 35B65 Key words Cross diffusion system; Uniform persistence; Global attractor Introduction In a recent work [16], we studied the global existence of a triangular cross diffusion parabolic systems of the type  ∂u   = ∇[(d1 + α11 u + α12 v)∇u + β11 u∇v] + u(a1 − b1 u − c1 v),  ∂t (1.1)   ∂v = ∇[(d + α u + α v)∇v] + v(a − b u − c v),  21 22 2 ∂t ∗ The author acknowledges the support by the NSF grant #DMS0305219, Applied Mathematics 509 510 D Le, T.T Nguyen which is supplied with the Neumann (r1 = r2 = 0) or Robin type boundary conditions ∂u + r1 (x)u = 0, ∂n ∂v + r2 (x)v = ∂n (1.2) on the boundary ∂Ω of a bounded domain Ω in IRn Here r1 , r2 are given nonnegative smooth functions on ∂Ω The initial conditions are described by u(x, 0) = u0 (x) and v(x, 0) = v0 (x), x ∈ Ω Here u0 , v0 ∈ W 1,p (Ω) for some p > n The system (1.1) has its origin from the Shigesada, Kawasaki and Teramoto model ([21])  ∂u  ′ ′  = ∆[(d1 + α11 u + α12 v)u] + u(a1 − b1 u − c1 v),  ∂t (1.3)  ∂v  ′ ′  = ∆[(d2 + α21 u + α22 v)v] + v(a2 − b2 u − c2 v), ∂t in population dynamics, which has been recently investigated to study the competition of ′ two species with cross diffusion effects In the context of ecology, di ’s and αij ’s are the self and cross dispersal rates, ’s represent growth rates, b1 , c2 denote self-limitation rates, and c1 , b2 are the interaction rates ′ = In this case, our Many works have been done under the assumption that α21 system (1.1) is a bit more general by having the term α21 u in the equation for v Furthermore, the flux components in (1.1), when α12 = β11 , not have to be gradients of some functions as described in (1.3) As far as we know, only global existence results were obtained for this system In particular, one can find global existence results for a simplified version of (1.3) (when ′ = 0) in [4, 13, 16, 19], and a regularity result for the full system in [15] α21 A central issue in population dynamics is the long-term development of populations, and one finds terms such as uniform persistence, coexistence, and extinction describing important special types of asymptotic behavior of the solutions of associated model equations If αij , βij are all zero, (1.1) reduced to the well known Lotka Volterra system, whose persistence property has been widely studied (see [8] for a good reference) However, to the best of our knowledge, this issue has not ever been addressed for cross diffusion cases This is, of course, due to the presence of the cross diffusion terms making necessary a priori estimates extremely difficult In our previous results [13] and [19], we proved the existence of the global attractor for the system (1.1) with α21 = Global existence results for the case α21 > were established in [16] Recently, in [17], we can only show that the L∞ norms of solutions of (1.1) are ultimately uniformly bounded We should remark that the presence of the term α21 u in the self-diffusion term in the equation of v makes the methods in [4, 13, 19] inapplicable Furthermore, these methods require that the dimension n is less than This restriction is not assumed in this current paper (and [16, 17]) Steady state or coexistence problems for similar systems were also extensively studied (see [10] and the reference therein) However, whether these coexistence states are observable, that is their stability, is still yet to be determined This question remains widely open even for the simpler Lotka-Volterra counterpart Coexistence in the sense of uniform persistence would then be more appropriate and realistic Roughly speaking, uniform persistence means that there are positive threshold levels below which time dependent solutions will 511 Cross diffusion parabolic systems never be for large t In biological terms, this means that no species will be either wiped out or completely invaded by others Persistence theories for general dynamical systems have been available for some years (see [9] and the references therein) It is now well known that the first step needed to apply these theories to a concrete model is to establish the existence of the global attractor For regular diffusion cases, by the smoothing effect of parabolic equations, this type of results is almost immediate as long as one can show that the L∞ norms of the solutions are ultimately uniformly bounded However, this is not the case for cross diffusion systems as one has to go further to show that the solutions are regular in higher norms, which are also uniformly bounded To achieve this, more sophisticated PDE techniques will be needed Our first main result is to obtain uniform estimates in higher norms to establish the existence of an absorbing ball in the W 1,p space as well as the compactness of the semiflow Since u, v are population densities, only positive solutions are of interest in this paper We then study the dynamics of the system on the positive cone of W 1,p , and prove the following theorem in Section Theorem 1.1 Assume that αij ≥ 0, di , β11 > 0, i, j = 1, and α11 > α21 , α22 > α12 , and α22 = α12 + β11 (1.4) Then (1.1) defines a dynamical system on W+1,p (Ω), the positive cone of W 1,p (Ω), for some p > n This dynamical system possesses a global attractor in W 1,p (Ω) Furthermore, there exist ν > and a positive constant C∞ independent of initial conditions such that (1.5) u(., t) C ν , v(., t) C ν ≤ C∞ for sufficiently large t In population dynamics terms, the first two conditions in (1.4) means that self diffusion rates are stronger than cross diffusion ones The third condition is a technical one In fact, this condition was only used in [16, 17] to derive uniform estimates for L∞ norms of the solutions via the existence of a Lyapunov function The proof in this paper employs Morrey’s estimates and imbedding theorems to achieve higher regularity Once again, we should point out that the techniques in [4], in the absence of the term α21 u, can only give that the W 1,p norms not blow up, and hence the global existence result Meanwhile, [19, 16] not provide uniform estimates like (1.5) for first order derivatives, which will be crucial for our proof of persistence below Moreover, our technique works for more general systems and requires only uniform L∞ estimates at the onset (see the assumptions (Q.1), (Q.2) in Section 2) Thus, Theorem 2.1 and Theorem 2.2 in Section can apply to much more general settings, provided that L∞ estimates are derived by other means Our next goal is to study the uniform persistence property of positive solutions of (1.1) We take advantage of the theory developed in [9] for dynamical systems (see Theorem 3.1), and apply it to our model We will prove the following result 512 D Le, T.T Nguyen Theorem 1.2 Assume (1.4) holds, and that the principal eigenvalues of the following problems: λψ = d1 ∆ψ + a1 ψ, ∂ψ + r1 ψ = 0, ∂n λφ = d2 ∆φ + a2 φ, ∂φ + r2 φ = 0, ∂n and (1.6) are positive Moreover, we assume that either (P.1) r1 = r2 ≡ and b1 a1 c1 > > , b2 a2 c2 or (P.2) r1 , r2 = and b1 α11 , b2 2α21 > a1 > max a2 c1 2α12 , c2 α22 , and (r.1) α12 > β11 and d1 α22 > 2d2 β11 ; (r.2) the quantities α21 , α12 − β11 , |a1 d2 − a2 d1 | and supx∈∂Ω |r1 (x) − r2 (x)| are all sufficiently small Then the system (1.1) is uniformly persistent, that is, there exists η > such that for any initial data u0 , v0 ∈ W 1,p (Ω) with u0 , v0 > we have lim inf u(., t) t→∞ C (Ω) ≥ η, lim inf v(., t) t→∞ C (Ω) ≥ η (1.7) In the context of biology, this means that no species is completely invaded or wiped out by the other so that they coexist in time From the structure of (1.1), the positivity of λ in (1.6) and the results of [2], it is known that the system possesses three trivial and semitrivial steady states (0, 0), (0, v∗ ) and (u∗ , 0) The trivial one describes the situation when both species are wiped out from the environment The other two semitrivial solutions model the survival of one species while the other is completely invaded The positivity of the principal eigenvalues in (1.6) gives the instability of the trivial steady state (see Proposition 3.1) Our conditions (P.1), (P.2) are essentially to guarantee that the two semitrivial steady states are unstable (or repelling) in their complement directions It is worth noticing that (P.1) is already well known for the Lotka-Volterra counterparts with homogeneous Neumann boundary conditions (see [2, 3, 8] and the references therein) It is not quite surprising to see that the cross diffusion parameters (αij , β11 ) not manifest in this case as the semitrivial steady states u∗ , v∗ are being just constants The situation will be more interesting when we consider (P.2) and the Robin boundary conditions in (1.1) Now, the semitrivial steady states are nonhomogeneous; and the cross diffusion (or gradient) effects will play an essential role 513 Cross diffusion parabolic systems The proof of this theorem will be presented at the end of Section In fact, we will establish sufficient conditions for the uniform persistence of each component That is to say when one species is not wiped out by the other (see Proposition 3.2 with Lemma 3.1, and Proposition 3.3 with Lemma 3.2) Finally, we would like to remark that the uniform persistence property in this paper is established in the C norm instead of the usual L∞ norm widely used in literature of Lotka-Volterra systems This is in part due to the setting of the phase space W 1,p for strongly coupled parabolic systems (see [1]) So, our persistence result does not rule out the possibility that solutions might form spikes at some points but approach zero almost everywhere as t → ∞ That type of behavior can be seen in some models for chemotaxis, which also involve a form of strong coupling, so it may be that the results presented here are optimal However, it is naturally to ask if it is impossible for one species can survive in the sense that its density is going to be almost negligible (that is, the L∞ norm goes to zero) while oscillating wildly to maintain the positivity of its C norm The answer to this question is still under investigation Estimates for the gradients In this section we will establish the uniform bound (1.5) for the gradients and prove Theorem 1.1 In fact, we will consider a more general parabolic system  ∂u   = ∇[P (u, v)∇u + R(u, v)∇v] + f (u, v),  ∂t (2.1)  ∂v   = ∇[Q(u, v)∇v] + g(u, v), ∂t with Neumann or Robin boundary conditions For the sake of simplicity, we will deal with ∂u ∂v the Neumann conditions = = in the proof below, and leave the Robin case to ∂n ∂n Remark 2.1 In order to prove (1.5) for (2.1), we assume the following conditions on the parameters of the system and the uniform boundedness of the solutions (Q.1) There exists a positive constant d such that P (u, v), Q(u, v) ≥ d Moreover, there is a constant C such that |R(u, v)| ≤ C|u| (Q.2) The solutions are uniformly bounded That is lim sup u(., t) t→∞ ∞ , lim sup t→∞ v(., t) ∞ ≤ C∞ (2.2) for some constant C∞ independent of the initial data u0 , v0 Indeed, we proved in [12] that weak bounded solutions of triangular parabolic systems including (2.1) are Hăolder continuous and therefore classical (see [1]) Moreover the C α norms of solutions are ultimately bounded by a positive constant dependent only on their 514 D Le, T.T Nguyen L∞ norms Thus, (2.2) implies the existence of a constant C∞ (α) independent of initial data such that lim sup u(., t) t→∞ C α , lim sup t→∞ v(., t) Cα ≤ C∞ (α), ∀α ∈ (0, 1) (2.3) Our main estimate of this section is the following Theorem 2.1 Let (u, v) be a nonnegative solution of (2.1) satisfying (Q.1), (Q.2) For any p ≥ 1, there exists a positive constant C∞,p independent of the initial data such that lim sup u(., t) 1,p + lim sup v(., t) 1,p ≤ C∞,p (2.4) t→∞ t→∞ Furthermore, the following stronger estimate also holds Theorem 2.2 Let (u, v) be a nonnegative solution of (2.1) satisfying (Q.1) and (Q.2) There exist finite constants C∞ and ν > such that lim sup u(., t) t→∞ Cν + lim sup v(., t) t→∞ Cν ≤ C∞ (2.5) The main idea of the proof is to use the imbedding results for Morrey’s spaces We recall the definitions of the Morrey space M p,λ (Ω) and the Sobolev-Morrey space W 1,(p,λ) Let BR (x) denotes a cube centered at x with radius R in IRn We say that f ∈ M p,λ (Ω) if f ∈ Lp (Ω) and f p M p,λ := ρ−λ sup x∈Ω,ρ>0 |f |p dy < ∞ Bρ (x) Moreover, f is in the Sobolev-Morrey space W 1,(p,λ) if f ∈ W 1,p (Ω) and f p W 1,(p,λ) := f p M p,λ + ∇f p M p,λ < ∞ p(n−λ) , we then have the following imbedding result If λ < n − p, p ≥ 1, and pλ = n−λ−p (see Theorem 2.5 in [5]) W 1,(p,λ) (B) ⊂ M pλ ,λ (B) (2.6) We then proceed by proving some estimates for the Morrey norms of the gradients of the solutions In the sequel, the temporal variable t is always assumed to be sufficiently large such that (see (2.3)) u(., t) Cα , v(., t) Cα ≤ C∞ (α), ∀α ∈ (0, 1) and t ≥ T, (2.7) where T may depend on the initial data From now on, let us fix a point (x, t) ∈ Ω × (T, ∞) As far as no ambiguity can arise, we write BR = BR (x), ΩR = Ω BR , and QR = ΩR × [t − R2 , t] We first have the following technical lemma 515 Cross diffusion parabolic systems Lemma 2.1 For sufficiently small R > 0, we have the following estimate (|∇u(x, t)|2 + |∇v(x, t)|2 ) dx+ [u2t + vt2 + |∆u|2 + |∆v|2 ] dz ≤ CRn−2+2α ΩR QR For the proof, we will need two useful results from [11] by Ladyzhenskaya et al for scalar functions It is easy to see that they also hold for vector valued functions as we restate in the following lemmas Lemma 2.2 ([11, Lemma II.5.4]) For any function u in W 1,2s+2 (Ω, IRm ) and any ∂u ξ vanishes on ∂Ω, we have smooth function ξ such that ∂n |∇u|2s+2 ξ dx ≤ Cosc2 {u, Ω} Ω (|∇u|2s−2 |∆u|2 ξ + |∇u|2s |∇ξ|2 ) dx (2.8) Ω Here, C is a constant depending on n, m, s Lemma 2.3 ([11, Lemma II.5.3]) Let α > and v be a nonnegative function such that for any ball BR and ΩR = Ω BR the estimate v(x) dx ≤ CRn−2+α ΩR holds Then for any function ξ from W01,2 (BR ) the inequality |∇ξ|2 dx v(x)ξ dx ≤ CRα (2.9) ΩR ΩR is valid Proof of Lemma 2.1 Let ξ(x, t) be a cut off function for QR and Q2R That is, ξ = on QR and ξ = outside Q2R Integration by parts in x gives vt ∆vξ dz = [− Q2R Q2R ∂(|∇v|2 ξ ) + |∇v|2 ξξt − vt ∇vξ∇ξ] dz ∂t We test the equation of v by −∆vξ Since ξ(x, t − 2R2 ) = 0, the above and a simple use of the Young inequality yield |∇v(x, t)|2 dx |∆v|2 ξ dz + ΩR Q2R [ǫvt2 ξ + C(|∇u|4 + |∇v|4 )ξ ] dz ≤ (2.10) Q2 R ξ dz |∇v|2 (|ξt | + |∇ξ|2 ) dz + C + C Q2R Q2R 516 D Le, T.T Nguyen Here, we have used the fact that f, g are uniformly bounded thanks to (2.2) Also, because the solutions are classical, the integrals of |∇u|4 , |∇v|4 make sense Similarly, test the equation of u by −∆uξ to get |∇u(x, t)|2 dx |∆u|2 ξ dz + ΩR Q2R u2t ξ dz + C ≤ ǫ (|∇u|4 + |∇v|4 + C|∆v|2 )ξ dz Q2 R Q2R ξ dz |∇u|2 (|ξt | + |∇ξ|2 ) dz + C + C (2.11) Q2R Q2R From the equations of (2.1), we also infer u2t + vt2 ≤ C(|∆u|2 + |∆v|2 + |∇u|4 + |∇v|4 + |∇u|2 + |∇v|2 + 1) (2.12) Using this in (2.10), (2.11) and adding them, we get (|∇u(x, t)|2 + |∇v(x, t)|2 ) dx (|∆u|2 + |∆v|2 )ξ dz + ΩR (2.13) Q2R (|∇u|2 + |∇v|2 )(ξ + |ξt | + |∇ξ|2 ) dz ≤ C Q2R ξ dz (|∇u|4 + |∇v|4 )ξ dz + C + C Q2R Q2R Using Lemma 2.2, we have (|∇u|4 + |∇v|4 )ξ dz ≤ CRα Q2R (|∆u|2 + |∆v|2 )ξ + (|∇u|2 + |∇v|2 )|∇ξ|2 ) dz Q2R Thus, for sufficiently small R, we see that the integrals of |∇u|4 , |∇v|4 in (2.13) can be absorbed to the left This shows that the quantity (|∆u|2 + |∆v|2 )ξ dz (|∇u|2 + |∇v|2 ) dx + ΩR Q2R can be majorized by [(|∇u|2 + |∇v|2 )(ξ + |ξt | + |∇ξ|2 ) + ξ ] dz C (2.14) Q2R This fact and (2.12), together with another use of Lemma 2.2, show that the quantity is also bounded by (2.14) (|∆u|2 + |∆v|2 ) dz (u2t + vt2 ) dz + (|∇u(x, t)|2 + |∇v(x, t)|2 ) dx + ΩR QR QR 517 Cross diffusion parabolic systems Finally, by testing equations of u and v in (2.1) with (u − uR )ξ and (v − vR )ξ respectively, with uR , vR being the averages of u, v over QR and ξ being the cut-off function for QR and Q3R , we can easily prove that (|∇u|2 + |∇v|2 ) dz ≤ CRn+2α Q2R Putting this and the fact that |ξt |, |∇ξ|2 ≤ CR−2 into (2.14), we see that the claims in our lemma are established The following lemma shows that ∇u, ∇v are uniformly bounded in W 1,(2,n−4+2α) (ΩR ) norms so that the imbedding (2.6) can be used Lemma 2.4 For R > sufficiently small, we have the following estimates : (u2t + vt2 ) dx ≤ CRn−4+2α , (2.15) (|∆u|2 + |∆v|2 ) dx ≤ CRn−4+2α (2.16) ΩR and ΩR Proof Again, let ξ(x, t) be a cut off function for QR and Q2R We now test the equation of v with −(vt ξ )t Integration by parts in t, x gives Q2R ∂(vt2 ξ ) dz ∂t (Q∇v)t ∇(vt ξ ) dz vt2 ξξt dz + − Q2R Q2R gt (u, v)vt ξ dz = (2.17) Q2R ∂v = on ∂Ω2R Note that, by the choice of ξ and the Neumann condition of v, ξ ∂n Therefore the boundary integrals resulting in the integration by parts are all zero As (Q∇v)t ∇(vt ξ ) = (Q∇vt + Qu ut ∇v + Qv vt ∇v)(∇vt ξ + 2vt ξ∇ξ), we easily see that (2.17), the ellipticity condition (Q.1), (2.2) and the facts that ξ(x, t − 2R2 ) = and |gt (u, v)| ≤ C|ut | + |vt | give ΩR vt2 dx |∇vt |2 ξ dz ≤ C + Q2R (u2t + vt2 )(ξ + |ξt |) dz + Q2R [|vt ∇vt ξ∇ξ| + |(|ut | + |vt |)∇v∇vt ξ | + |(|ut | + |vt |)∇vvt ξ∇ξ|] dz + C Q2R 518 D Le, T.T Nguyen Using Young’s inequality, we have |vt ∇vt ξ∇ξ| |(ut + vt )∇v∇vt ξ | |(ut + vt )∇vvt ξ∇ξ| ≤ ǫ|∇vt |2 ξ + C(ε)vt2 |∇ξ|2 , ≤ ǫ|∇vt |2 ξ + C(ε)(u2t + vt2 )|∇v|2 ξ , ≤ C[(u2t + vt2 )|∇v|2 ξ + vt2 |∇ξ|2 ] Therefore, ΩR vt2 dx |∇vt |2 ξ dz + (2.18) Q2R (u2t + vt2 )(ξ + |ξt | + |∇ξ|2 ) dz (u2t + vt2 )|∇v|2 ξ dz + C ≤ C Q2R Q2R Next, we test the equation of u with −(ut ξ )t and note that (R∇v)t ∇(ut ξ ) = (Ru ut ∇v + R∇vt )(∇ut ξ + 2ut ξ∇ξ) By Hăolder inequality, we also have |(Rv)t (ut )| dz |∇ut |2 ξ dz + C ≤ ǫ |∇vt |2 ξ dz Q2R Q2R Q2R (u2t |∇v|2 ξ + u2t |∇ξ|2 ) dz + Q2R Hence, arguing similarly as before, we also have ΩR u2t dx |∇vt |2 ξ dz |∇ut |2 ξ dz ≤ C + (2.19) Q2R Q2R (u2t + vt2 )(|∇u|2 + |∇v|2 )ξ + (u2t + vt2 )(ξ + |ξt | + |∇ξ|2 ) dz + C Q2R By (2.18), the integral of |∇vt |2 ξ can be eliminated from the right hand side The result and (2.18) together show that ΩR u2t + vt2 dx (|∇ut |2 + |∇vt |2 )ξ dz ≤ + (2.20) Q2R (u2t + vt2 )(|∇u|2 + |∇v|2 )ξ + (u2t + vt2 )(ξ + |ξt | + |∇ξ|2 ) dz + C Q2R (|∇u|2 + |∇v|2 ) dx ≤ cRn−2+2α This allows us As we proved in Lemma 2.1, ΩR to apply Lemma 2.3, with the function v replaced by |∇u|2 + |∇v|2 , to derive (|∇u|2 + |∇v|2 )vt2 ξ dz ≤ cR2α Q2R [|∇vt |2 ξ + vt2 |∇ξ|2 ] dz Q2R 519 Cross diffusion parabolic systems and (|∇u|2 + |∇v|2 )u2t ξ dz ≤ cR2α Q2R [|∇ut |2 ξ + u2t |∇ξ|2 ] dz Q2R Hence, for R sufficiently small, we obtain from (2.20) that ΩR u2t (x, t) + vt2 (x, t) dx ≤ C (u2t + vt2 )(ξ + |ξt | + |∇ξ|2 ) dz (2.21) Q2R Applying Lemma 2.1 and using the fact that |ξt |, |∇ξ|2 ≤ CR−2 , we obtain the desired inequality (2.15) For (2.16), we solve ∆u and ∆v in terms of ut , vt , ∇u, and ∇v and then integrate them over ΩR to get (|∆u|2 + |∆v|2 )ξ dx ≤ C ΩR ΩR (u2t + vt2 + |∇u|2 + |∇v|2 + |∇u|4 + |∇v|4 + 1)ξ dx We then use Lemma 2.2 again to absorb the term |∇u|4 + |∇v|4 to the left hand side The result is (|∆u|2 + |∆v|2 )ξ dx ≤ C ΩR ΩR [(u2t + vt2 )ξ + (|∇u|2 + |∇v|2 )(ξ + |∇ξ|2 ) + ξ ] dx This, Lemma 2.1 and (2.15) give (2.16), and complete our proof We are now ready to give Proof of Theorem 2.1 Thanks to the above lemmas, the estimate ΩR [(u2t + vt2 ) + (|∇u|2 + |∇v|2 ) + (|∇u|4 + |∇v|4 ) + (|∆u|2 + |∆v|2 )] dx ≤ CRn−4+2α holds for some constant C independent of the initial data if t is sufficiently large By rewriting the equations of u, v as P ∆u = F and Q∆v = G, with F , G depending on the first order derivatives of u, v in x, t, and using the above estimates, we can apply [20, Lemma 4.1] to assert that the norms of ∇u and ∇v in W 1,(2,λ) (ΩR ), with λ = n − + 2α, are uniformly bounded Therefore, by the imbedding (2.6), and the fact that M 2λ ,λ ⊂ L2λ , we have ∇u(•, t) L2λ (Ω) and ∇v(•, t) L2λ (Ω) , with 2λ = 2(4−2α) 2−2α , are bounded by some constant C Since α can be arbitrarily chosen in (0, 1), 2λ can be as large as desired This proves (2.4) Regarding (2.5), we rewrite the equation of v as follows: vt = Q∆v + G with G = Qu ∇u∇v + Qv |∇v|2 + g Since u, v are Hăolder continuous with uniformly bounded norms, we can regard Q as a Hăolder continuous function in (x, t) Therefore, we can apply ii) of [13, Lemma 2.5] here to obtain 520 D Le, T.T Nguyen t v(., t) Cµ ≤ t−β e−δt v(., τ ) r (t − s)−β e−δ(t−s) ( ∇u + τ 2r + ∇v 2r + g r )ds (2.22) for any < τ < t and 2β > µ + n/r Using (2.4) and (2.2), for sufficiently large t, τ , we have t v(., t) Cµ ≤ C(r)t−β e−δt + C(r) (t − s)−β e−δ(t−s) ds (2.23) τ for some constant C(r) independent of the initial data The above integral is finite for all t if β ∈ (0, 1) Obviously, we can choose r sufficiently large and µ > such that β < 1, and therefore prove that v(., t) C µ is uniformly bounded for large t Finally, such Hăolder estimate for v allows us to follow the proof of Theorem 2.2 in [13] to get the uniform estimate for u(., t) C µ as desired Remark 2.1 The case of Robin boundary conditions can be reduced to the Neumann one by a simple change of variables First of all, since our proof based on the local estimates of Lemma 2.1 and Lemma 2.4, we need only to study these inequalities near the boundary As ∂Ω is smooth, we can locally flatten the boundary and assume that ∂Ω is the plane {xn = 0} Furthermore, we can take ΩR = {(x′ , xn ) : xn > 0, |(x′ , xn )| < r} The boundary conditions become ∂u + r1 (x′ )u = 0, ∂xn ∂v + r2 (x′ )v = ∂xn We then introduce U (x′ , xn ) = exp(xn r1 (x′ ))u(x′ , xn ), V (x′ , xn ) = exp(xn r2 (x′ ))v(x′ , xn ) Obviously, U, V satisfy the Neumann boundary condition on xn = Simple calculations also show that U, V verify a system similar to that for u, v, and the conditions (Q.1), (Q.2) are still valid In fact, there will be some extra terms occurring in the divergence parts of the equations for U, V , but these terms can be handled by a simple use of Young’s inequality so that our proof can go on with minor modifications Thus Theorem 2.1 applies to U, V , and the estimates for u, v then follow We conclude this section by giving the proof of Theorem 1.1 Proof of Theorem 1.1: In our recent works (see [16], [17]), we proved that nonnegative weak solutions of (1.1) are ultimately uniformly bounded in their L∞ norms Therefore, the conditions (Q.1), (Q.2) are verified by (1.1), and Theorem 2.1 applies here The estimate (2.4) asserts the existence of an absorbing ball in W 1,p (Ω) attracting all solutions The compactness of associated semiflow in W 1,p (Ω) comes from the estimate (1.5) The existence of the global attractor then follows (see [7]) 521 Cross diffusion parabolic systems Persistence results In this section we shall consider the question of persistence and prove Theorem 1.2 Our proof mainly bases on a persistence result in [9] for general dynamical systems defined on metric spaces In order to restate this result, let us first recall some definitions in the dynamical system theory Let (X, d) be a metric space and Φ be a semiflow on X A subset A ⊂ X is said to be an attractor for Φ if A is nonempty, compact, invariant, and there exists some open neighborhood U of A in X such that limt→∞ d(Φt (u), A) = for all u ∈ U Here, d(x, A) is the usual Hausdorff distance from x to the set A If A is an attractor which attracts every point in X, A is called global attractor For a nonempty invariant set M , the set W s (M ) := {x ∈ X : limt→∞ d(Φt (x), M ) = 0} is called the stable set of M A nonempty invariant subset M of X is said to be isolated if it is the maximal invariant set in some neighborhood of itself Let A and B be two isolated invariant sets A is said to be chained to B, denoted by A → B, if there exists a globally defined trajectory Φt (x), t ∈ (−∞, ∞), through some x ∈ A B whose range has compact closure such that the omega limit set ω(x) ⊂ B and the alpha limit set α(x) ⊂ A A finite sequence {M1 , M2 , , Mk } of isolated invariant sets is called a chain if M1 → M2 → → Mk The chain is called a cycle if Mk = M1 Let X0 ⊂ X be an open set and ∂X0 = X \ X0 Assume that X0 is positively invariant Let p(x) = d(x, ∂X0 ), the distance from x to ∂X0 Φ is said to be uniformly persistent with respect to (X0 , ∂X0 , p) if there exists η > such that lim inf p(Φt (x)) ≥ η t→∞ for all x ∈ X0 The following uniform persistence result is established in [9] Theorem 3.1 (Theorem 4.3 in [9]) Assume that (C1) Φ has a global attractor A; (C2) There exists a finite sequence M = {M1 , , Mk } of pairwise disjoint, compact and isolated invariant sets in ∂X0 with the following properties: (m.1) x∈∂X0 ω(x) ⊂ k i=1 Mi , (m.2) no set of M forms a cycle in ∂X0 , (m.3) Mi is isolated in X, (m.4) W s (Mi ) X0 = ∅ for each i = 1, , k Then there exists δ > such that for any x ∈ X0 , the following inequality holds inf d(y, ∂X0 ) > δ y∈ω(x) We will apply this theorem to obtain the uniform persistence for the system (1.1) Let (Ω) = {u ∈ C (Ω) : u(x) ≥ ∀x ∈ Ω} We define C+ 1 X = C+ (Ω) × C+ (Ω) and X0 = {(u, v) ∈ X : u(x) > and v(x) > 0}, 522 D Le, T.T Nguyen with norm (u, v) X = u C (Ω) + v C (Ω) Thanks to Theorem 1.1, the solutions with initial data in W 1,p (Ω) will become C smooth so that we need only to consider the system (1.1) and its associated semiflow Φ on X That is, for any (u0 , v0 ) in X, let (u(•, t), v(•, t)) be the solution to (1.1) and define Φt (u0 , v0 ) = (u(•, t), v(•, t)) for all t ≥ Firstly, we will see that X0 is positively invariant with respect to Φ Indeed, let us rewrite the equation of u as follows: ut = div(a(x, t)∇u + b(x, t)u) + c(x, t), where a(x, t) = P (u(x, t), v(x, t)) ≥ d > 0, b(x, t) = Ru ∇v(x, t) and c(x, t) = f (u(x, t), v(x, t)) Here and throughout this section, we denote P (u, v) = d1 + α11 u + α12 v, R(u) = β11 u, Q(u, v) = d2 + α21 u + α22 v, and f (u, v) = u(a1 − b1 u − c1 v), g(u, v) = v(a2 − b2 u − c2 v) By virtue of Theorem 1.1, we see that a, b, c are bounded Using the strong positivity result in [18], we see that u(x, t) > for all t Similar argument shows that the component v also stays positive Hence, Φt (u0 , v0 ) ∈ X0 for all t Theorem 1.1 also asserts that Φ has a global attractor in X, and thus (C1) is verified Next, we consider the condition (C.2) It is clear that the “boundary” parts u = or v = of X0 are also invariant with respect to Φ On these boundaries, the dynamics of (1.1) is reduced to those of the following scalar parabolic equations ut vt = ∇(P (u, 0)∇u) + f (u, 0), u(0) > 0, = ∇(Q(0, v)∇v) + g(0, v), v(0) > (3.1) (3.2) Investigating the dynamics of these equations leads us to the following steady state equations ∇(P (u∗ , 0)∇u∗ ) + f (u∗ , 0) = 0, ∇(Q(0, v∗ )∇v∗ ) + g(0, v∗ ) = 0, together with the boundary conditions as in (1.2) If the principal eigenvalues of (1.6) are positive, the above equations admit unique solutions, which are denoted respectively by u∗ and v∗ Furthermore, the solutions u(x, t), v(x, t) of (3.1), (3.2) converge to u∗ , v∗ , respectively, in the C(Ω) norm as t tends to infinity Meanwhile, the trivial solution is an unstable steady state for both equations These claims are obtained by following closely the proof of [2, Corollary 2.4] or [3, Theorem 1.2], where the Dirichlet boundary condition was assumed Therefore, the sets M0 = (0, 0), M1 = (u∗ , 0), and M2 = (0, v∗ ) are pairwise disjoint, compact and isolated invariant sets in ∂X0 with respect to Φ Moreover, no set of {Mi } can form a cycle in ∂X0 ; and x∈∂X0 ω(x) ⊂ i=0 Mi We thus show that the conditions (m.1) and (m.2) are satisfied Checking (m.3) and (m.4) requires much more effort The role of the parameters r1 , r2 will play an important role here Let us assume that the system (1.1) satisfies the Robin 523 Cross diffusion parabolic systems boundary condition (1.2) with r1 , r2 = The Neumann case is simpler, and will be discussed later in Remark 3.1 We discuss first the property (m.4) at M0 We will show below that the instability of M0 is determined by the principal eigenvalue λ of (see (1.6)) λφ = d2 ∆φ + a2 φ, ∂φ + r2 φ = ∂n (3.3) Proposition 3.1 Assume that the principal eigenvalue λ of (3.3) is positive There exists η0 > such that for any solution (u, v) of (1.1) with (u0 , v0 ) ∈ X0 , we have lim sup (u(., t), v(., t)) t→∞ X ≥ η0 Proof Let φ be the positive eigenfunction associated to the principal eigenvalue λ of (3.3) By testing the equation of v by φ and (3.3) by v, we subtract the results to get d dt [−Q0 ∇v∇φ + (g − a2 v)φ] dx − vφ dx + vφ dx = λ Ω Ω Ω ∂Ω Q0 r2 vφ dσ (3.4) Here, we denoted Q0 = Q − d2 = α21 u + α22 v Integration by parts yields v∇(Q0 ∇φ) dx + Q0 ∇v∇φ dx = − Ω Ω ∂Ω r2 Q0 vφ dσ Putting this in (3.4), we infer d dt Ω vφ vφ dx + vφ dx = λ Ω Ω ∇(Q0 ∇φ) dx − φ (b2 u + c2 v)vφ dx Ω Now, suppose that our claim was false For any η > 0, there would be a solution u, v such that (u(., t), v(., t)) X ≤ η when t is large This implies that the quantities |∇(Q0 ∇φ)| and (b2 u + c2 v) can be very small Thus, if η is sufficiently small, then the φ above equation yields d λ vφ dx ≥ vφ dx dt Ω Ω This shows that, as t → ∞, (u, v) v(., t)φ dx goes to infinity, contradicting the fact that Ω X is bounded Our proof is complete Next, we study M1 and M2 Our main assumption for (m.3) and (m.4) to hold is the instability of M1 , M2 in their complement v, u directions, respectively To this end, we consider the linearization of the system (1.1) at a general steady state point (u, v) λψ = ∇[(Pu ψ + Pv φ)∇u + P ∇ψ + (Ru ψ + Rv φ)∇v + R∇φ] + fu ψ + fv φ, λφ = ∇[(Qu ψ + Qv φ)∇v + Q∇φ] + gu ψ + gv φ (3.5) 524 D Le, T.T Nguyen Here ψ (respectively, φ) satisfies the boundary condition of u (respectively, v) in (1.2) Putting (u, v) = (0, v∗ ) and (ψ, φ) = (ψ, 0), the instability of M2 = (0, v∗ ) in the direction u is determined by the sign of the principal eigenvalue of the following system λψ = ∇(P (0, v∗ )∇ψ + Ru ψ∇v∗ ) + fu (0, v∗ )ψ, (3.6) with v∗ being the solution of = ∇(Q(0, v∗ )∇v∗ ) + g(0, v∗ ) (3.7) We shall establish the following repelling property of (0, v∗ ) Proposition 3.2 Suppose that the principal eigenvalue λ of (3.6) is positive If Pv − Ru = α12 − β11 is positive and sufficiently small, then there exists η0 > such that for any solution (u, v) of (1.1) with (u0 , v0 ) ∈ X0 , we have lim sup (u(., t), v(., t)) − (0, v∗ ) t→∞ ≥ η0 X Similarly, the instability of M1 = (u∗ , 0) in the direction v is determined by the sign of the principal eigenvalue of the following system λφ = ∇(Q(u∗ , 0)∇φ) + gv (u∗ , 0)φ, (3.8) with u∗ being the solution of = ∇(P (u∗ , 0)∇u∗ ) + f (u∗ , 0) (3.9) Proposition 3.3 Suppose that the principal eigenvalue λ of (3.8) is positive If Qu = α21 is positive and sufficiently small, then there exists η0 > such that for any solution (u, v) of (1.1) with (u0 , v0 ) ∈ X0 , we have lim sup (u(., t), v(., t)) − (u∗ , 0) t→∞ X ≥ η0 An immediate consequence of these propositions is that W s (Mi ) X0 = ∅, i = 0, 1, respectively Otherwise, by the definition of W s (Mi ), there exists (u0 , v0 ) ∈ X0 such that d((u(t), v(t)), Mi ) → as t → ∞, a contradiction to the above corresponding propositions Moreover, we also see that Mi is isolated in X Indeed, consider a neighborhood of Mi in X0 , V = {(u, v) ∈ X0 : d((u, v), Mi ) < η0 /2} For any (u0 , v0 ) ∈ X0 V , the above proposition shows that (u(t), v(t)) will inevitably exits V This means Mi is maximal in V , and isolated in X We now give the proof of Proposition 3.2 and Proposition 3.3 Proof of Proposition 3.2 The proof is by contradiction Assume that for any η > there exists a solution (u, v) of (1.1) and T > such that u(., t) C (Ω) , v(., t) − v∗ C (Ω) T Hereafter, we always consider t > T We denote P0 = P (0, v∗ ) and recall (3.5): λψ = ∇(P0 ∇ψ + Ru ψ∇v∗ ) + fu (0, v∗ )ψ u Set P (u, v) = P (s, v)ds We note that ∇P (u, v) = P ∇u + Pv u∇v Testing the above equation with P , we obtain = − ψP (u, v) dx λ P0 P ∇ψ∇u dx − Ω Ω P0 Pv u∇ψ∇v dx Ω Ru ψ∇v∗ (P ∇u + Pv u∇v) dx + − + fu ((0, v∗ ))ψP dx Ω Ω (P0 ∂Ω ∂ψ ∂v∗ + Ru ψ )P dσ ∂n ∂n (3.11) Similarly, we test the equation of u in (1.1) with P0 ψ (∇(P0 ψ) = Pv ψ∇v∗ + P0 ∇ψ), and get ∂ ∂t P P0 ∇u∇ψ dx − = − P0 uψ dx P Pv ψ∇u∇v∗ dx Ω Ω Ω R∇v(P0 ∇ψ + Pv ψ∇v∗ ) dx + − + ∂Ω f P0 ψ dx Ω Ω (P ∂u ∂v + R )P0 ψ dσ ∂n ∂n (3.12) From (3.11), (3.12) and the boundary condition (1.2), we find ∂ ∂t P0 uψ dx [P0 u∇v∇ψ − P ψ∇u∇v∗ ] dx ψP dx + (Pv − Ru ) = λ Ω Ω Ω (f P0 − fu (0, v∗ )P )ψ dx + I∂ , + (3.13) Ω (P − P u)r1 ψP0 + (P v∗ − uvP0 )Ru r2 ψ dσ ∂Ω Next, we shall show that the integrals on the right of (3.13) are either nonnegative or controlled by the first integral From the definition of the parameters, we have where I∂ = Pψ = (f P0 − fu (0, v∗ )P )ψ = α11 u2 ψ ≥ P0 uψ + Pv (v − v∗ )uψ, α11 u (c1 (v∗ − v) − b1 u)P0 uψ + fu (α12 (v∗ − v) − )uψ (d1 + α12 v)uψ + Hence, if η in (3.10) is sufficiently small, the above gives P ψ dx ≥ Ω (f P0 − fu (0, v∗ )P )ψ dx ≤ P0 uψ dx, Ω Ω λ P0 uψ dx Ω 526 D Le, T.T Nguyen On the other hand, integrate by parts to get − u∇(P ψ∇v∗ ) dx − P ψ∇u∇v∗ dx = Ω Ω = Ω uP ψ ∂v∗ dσ ∂n ∂Ω ∇(P ψ∇v∗ ) uP ψr2 v∗ dσ dx + uψ ψ ∂Ω ¯ the quantities |∇v|| ∇ψ |, Thanks to (3.10) and the fact that ψ > on Ω, ψ bounded Thus, if Pv − Ru is positive and sufficiently small, then [P0 u∇v∇ψ − P ψ∇u∇v∗ ] dx ≥ − (Pv −Ru ) Ω λ ∇(P ψ∇v∗ ) P0 ψ P0 uψ dx+(Pv −Ru ) Ω ∂Ω are P r2 v∗ uψ dσ Putting these facts in (3.13), we derive ∂ ∂t P0 uψ dx ≥ Ω λ P0 uψ dx + I∂ + (Pv − Ru ) Ω ∂Ω uP ψr2 v∗ dσ Finally, we study the boundary integrals Straightforward calculations show I∂ = ∂Ω [− α11 α11 ur1 P0 + (d1 (v∗ − v) + uv∗ )Ru r2 ]uψ dσ 2 If η in (3.10) is small, then it is clear that the quantity in the brackets can be very small Thus, I∂ can be controlled by the positive boundary integral (Pv − Ru ) Therefore ∂ ∂t λ P0 uψ dx ≥ Ω ∂Ω P0 uψ dx P r2 v∗ uψ dσ (3.14) Ω P (0, v∗ )uψ dx goes to infinity as t does This contradicts As λ > 0, this shows that Ω (3.10) and completes this proof Proof of Proposition 3.3 Along the line of the proof of Proposition 3.2, we assume that for any η > there exists a solution (u, v) of (1.1) and T > such that v(., t) C (Ω) + u(., t) − u∗ C (Ω) < η, for any t > T (3.15) Consider the equation (3.8) λφ = ∇(Q0 ∇φ) + gv (u∗ , 0)φ, v where Q0 = Q(u∗ , 0) Set Q(u, v) = Q(u, s)ds Test the above equation with Q and the equation of v in (1.1) with Q0 φ, we easily derive ∂ ∂t Q0 v∇u∇φ dx − Qu φQ dx + Qu Q0 vφ dx = λ Ω Ω Ω (gQ0 − gv (u∗ , 0)Q)φ dx + + Ω Qφ∇v∇u∗ dx Ω ∂Ω (Q − Qv)r2 φQ0 dσ.(3.16) 527 Cross diffusion parabolic systems First, for sufficiently small η in (3.15), it is not difficult to see that (gQ0 − gv (u∗ , 0)Q)φ dx ≤ Ω λ φQ dx Ω Next, by integration by parts and the boundary condition of u∗ , we get − Qφ∇v∇u∗ dx = v∇(Qφ∇u∗ ) dx + Qvφr1 u∗ dσ ∂Ω As |∇u||∇φ|/|φ| and |∇(Qφ∇u∗ )|/φ are bounded, if Qu is sufficiently small, we also have Ω Ω v∇(Qφ∇u∗ ) dx ≥ − Q0 v∇u∇φ dx and Qu Qu Ω Ω λ Q0 vφ dx Ω Put these estimates in (3.16) to see that ∂ ∂t Q0 vφ dx ≥ Ω λ Q0 vφ dx + Ω ∂Ω (Q − Qv)r2 φQ0 dσ + Qu ∂Ω Qvφr1 u∗ dσ (3.17) Concerning the boundary integrals, we note that Q − Qv = − α222 v Therefore, if η in (3.15) is sufficiently small, then the sum of the boundary integral is positive We then conclude that λ ∂ Q0 vφ dx ≥ Q0 vφ dx ∂t Ω Ω Q0 vφ dx tends to infinity, contradicting (3.15) Our proof is This inequality shows Ω complete Remark 3.1 If the boundary conditions are of Neumann type, then u∗ , v∗ , ψ, φ in the above proofs are just constant functions and our calculations will be much simpler In fact, it is easy to see that the smallness condition for Pv − Ru (respectively Qu ) in Proposition 3.2 (respectively Proposition 3.3) is no longer needed Next, we will present explicit and simple criteria on the parameters of (1.1) for the positivity of the principal eigenvalues of (3.6), (3.8) Lemma 3.1 Assume that either r1 = r2 ≡ and a1 /a2 > c1 /c2 , or a1 > max a2 c1 2α12 , c2 α22 , (3.18) and a) α12 > β11 ; b) d1 α22 ≥ 2d2 β11 ; supx∈∂Ω (r1 (x) − r2 (x))+ and (a2 d1 − a1 d2 )+ are sufficiently small 528 D Le, T.T Nguyen Then λ in (3.6) is positive Proof Set P0 = P (0, v∗ ), Q0 = Q(0, v∗ ) We test (3.6) with Q = (3.7) with P (0, v∗ )ψ Together, we get ψQ dx = − λ + = ∂Ω Ω (P0 ∂v∗ ∂ψ + Ru ψ )Q dσ + ∂n ∂n Q0 ψ|∇v∗ |2 dx + (Pv − Ru ) ∂Ω (P0 fu ψQ dx Ω (fu Q − g0 P0 )ψ dx Ω Ω + Q(0, s)ds and test Ru Q0 ψ|∇v∗ |2 dx P0 Q0 ∇ψ∇v∗ dx − Ω Ω v∗ ∂ψ ∂v∗ ∂v∗ Q0 P0 ψ + Ru ψ )Q dσ − dσ ∂n ∂n ∂n ∂Ω We need only show that the right hand side is positive Since Pv = α12 > β11 = Ru , the first term on the right is nonnegative For the second integral, we note that fu Q − g0 P0 = v∗ [(a1 − c1 v∗ )(d2 + α22 v∗ ) − (a2 − c2 v∗ )(d1 + α12 v∗ )] We study the quantity in the brackets by considering the quadratic F (X) = = α22 X) − (a2 − c2 X)(d1 + α12 X) 1 (c2 α12 − c1 α22 )X + ( a1 α22 − a2 α12 + c2 d1 − c1 d2 )X + a1 d2 − a2 d1 2 (a1 − c1 X)(d2 + First of all, by a simple use of maximum principles, we can easily show that < v∗ (x) ≤ a2 /c2 for all x ∈ Ω Let µ = inf Ω v∗ (x) > We will show that F (v∗ ) > Firstly, due to (3.18), F (0) = a1 d2 − a2 d1 and F (a2 /c2 ) = (a1 − a2 c1 a2 α22 )(d2 + ) > c2 2c2 Consider the case when the coefficient of X in F (X) is negative If F (0) ≥ then F (v∗ ) > because < µ ≤ v∗ (x) ≤ a2 /c2 If F (0) < 0, then F (X) = has two positive roots X1 , X2 with X2 > a2 /c2 Hence, if |F (0)| is sufficiently small then µ > X1 and therefore F (v∗ ) > Otherwise, by (3.18), we have F (v∗ ) ≥ ( 21 a1 α22 − a2 α12 + c2 d1 − c1 d2 )v∗ + F (0) If (c2 d1 − c1 d2 ) ≥ 0, the last quantity is obviously positive when either F (0) ≥ or F (0) < but |F (0)| is small Or else, because v∗ ≤ a2 /c2 we have F (v∗ ) ≥ (c2 d1 − c1 d2 ) a2 a1 c1 + a1 d2 − a2 d1 = a2 d2 ( − ) > c2 a2 c2 In all cases, F (v∗ ) > Thus, the second integral is also positive It remains to consider the boundary integrals In view of (1.2), they are ∂Ω (r2 − r1 )P0 Qψ dσ + ∂Ω r2 ψv∗2 α22 α22 d1 − β11 d2 + (Pv − Ru )v∗ 2 dσ 529 Cross diffusion parabolic systems The last integrand is positive due to the first condition in b) Therefore the above sum is nonnegative if either r2 ≥ r1 or r1 − r2 > but sufficiently small Therefore, under the stated assumptions in the lemma, λ is positive Similarly, we have the following result Lemma 3.2 Assume that either r1 = r2 ≡ and a2 /a1 > b2 /b1 , or a2 > max a1 b2 2α21 , b1 α11 , (3.19) and (a2 d1 − a1 d2 )− , supx∈∂Ω (r2 (x) − r1 (x))+ are sufficiently small Then λ in (3.8) is positive Proof Following the previous proof, we test (3.8) with P = Q0 φ (Q0 = Q(u∗ , 0), P0 = P (u∗ , 0)), to get [Qu P0 φ|∇u∗ |2 + (gv P − f Q0 )φ] dx+ φP dx = λ Ω Ω We note that r1 P0 u∗ − r2 P = (r1 − r2 )P0 u∗ + gv P − f Q0 = (a2 − b2 u∗ )(d1 + ∂Ω Pu 2 r2 u ∗ , u∗ P (s, 0)ds, (3.9) with (r1 P0 u∗ − r2 P )Q0 φ dσ and α11 u∗ )u∗ − (a1 − b1 u∗ )(d2 + α21 u∗ )u∗ The proof is then similar to that of Lemma 3.1 We omit the details We conclude this paper by giving the proof of Theorem 1.2 Proof of Theorem 1.2 It is clear that the stated conditions (P.1) or (P.2) satisfy those of our propositions and lemmas of this section The theorem then follows from Theorem 3.1 References [1] H Amann, Dynamic theory of quasilinear parabolic systems-III global existence, Math Z 202 (1989), pp 219-250 [2] R S Cantrell and C Cosner, Diffusive logistic equations with indefinite weights: 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Annali di Matematica Pura ed Applicata [16] D Le and T Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension, Proc Amer Math Soc 133, no 7,(2005),... A) = for all u ∈ U Here, d(x, A) is the usual Hausdorff distance from x to the set A If A is an attractor which attracts every point in X, A is called global attractor For a nonempty invariant

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